A general measure of the comparative judgments’ goodness in building up decision matrices within the AHP is consistency, measured through the random index. Indeed, for a consistent, positive, reciprocal matrix, every element is the ratio of the correspondent components in the eigenvector associated with the highest correspondent eigenvalue, thus allowing a perfect match for the interpretation of each element in a decision matrix as being a comparison-expressed as a ratio-of the criterions’ weights in the associated priority vector. Starting from Saaty’s famous examples in which the decision matrices with small random indices were showing an almost perfect fit between the estimated priority vector and the real ones, this algebraically general construction was almost unanimously recognized as an ideal for every decision matrix. A quest for improving consistency in every decision matrix, either through the achievement of consensus or through other methods, led to the development of two main directions, linguistic and bold consistency. This paper is raising two questions. First is about how relevant are Saaty’s famous examples. It is thus illustrated that in term of the relationship between the consistency ratio and the standard deviation of the associated priority vector, achieving consensus in these consecrated examples were highly improbable events. Second, following the spirit of Saaty’s famous examples, some experiments in which the true priority vectors are known are replicated on a larger scale, in order to see whether improving on the consistency is indeed conducing to a closer match between the priority vector and the true one. Results achieved so far revealed the fact that only in approximate one half of the decisions matrices, the effort of improving consistency (either linguistic or bold) led to a priority vector closer to the true one.